Following the approach by [1]} and [2]}, we describe the flow of each phase in the transition region, referred to as the dynamic meniscus, that connects the films of constant thickness downstream to the static meniscus upstream.
For each liquid, the vertical extent of the static meniscus is assumed much larger than the corresponding film thickness, allowing us to apply the lubrication theory to the flow.
In this steady lubrication approach, the momentum conservation equations in the \(z\) -direction read:
\(0 & = & -\frac{\partial p_1}{\partial z} + \mu _1 \frac{\partial ^2 u_{1}}{\partial x^2} - \rho _1 g,\qquad \qquad (0 < x < h_1(z), \quad z > 0) \\0 & = & -\frac{\partial p_2}{\partial z} + \mu _2 \frac{\partial ^2 u_{2}}{\partial x^2} - \rho _2 g.\qquad \qquad (h_1(z) < x < h_2(z), \quad z > \Delta H)\) 
